To determine which option represents a 3rd-degree binomial with a constant term of 8, we need to break down each option and identify its properties.
1. Understanding the terms:
- 3rd-degree: This means the highest power of the variable (commonly [tex]\(x\)[/tex]) is 3. In polynomial terms, this is called a cubic polynomial.
- Binomial: A polynomial that contains exactly two terms.
- Constant term: This is the term in the polynomial that does not contain the variable [tex]\(x\)[/tex]. It is a standalone number.
2. Analyzing each choice:
- Option (A): [tex]\(x^3 - x^2 + 8\)[/tex]
- This contains three terms ([tex]\(x^3\)[/tex], [tex]\(-x^2\)[/tex], and [tex]\(8\)[/tex]), so it is not a binomial. It is a trinomial.
- It does have a 3rd-degree term ([tex]\(x^3\)[/tex]).
- The constant term is [tex]\(8\)[/tex].
- However, since it is not a binomial, it does not meet all the criteria.
- Option (B): [tex]\(2x^8 + 3\)[/tex]
- This contains two terms ([tex]\(2x^8\)[/tex] and [tex]\(3\)[/tex]), so it is a binomial.
- The highest degree is 8 ([tex]\(2x^8\)[/tex]), so it is an 8th-degree polynomial.
- The constant term is [tex]\(3\)[/tex].
- This does not meet the criteria for being 3rd-degree.
- Option (C): [tex]\(-5x^3 + 8\)[/tex]
- This contains two terms ([tex]\(-5x^3\)[/tex] and [tex]\(8\)[/tex]), so it is a binomial.
- The highest degree is 3 ([tex]\(-5x^3\)[/tex]).
- The constant term is [tex]\(8\)[/tex].
- This meets all the criteria: it's a binomial, 3rd-degree, and has a constant term of [tex]\(8\)[/tex].
- Option (D): [tex]\(8x^3 + 2x + 3\)[/tex]
- This contains three terms ([tex]\(8x^3\)[/tex], [tex]\(2x\)[/tex], and [tex]\(3\)[/tex]), so it is not a binomial. It is a trinomial.
- The highest degree is 3 ([tex]\(8x^3\)[/tex]).
- The constant term is [tex]\(3\)[/tex].
- However, since it is not a binomial, it does not meet all the criteria.
Based on the analysis, the correct answer is:
(C) [tex]\(-5x^3 + 8\)[/tex]
